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Proof of Sequences: Orders and Representations

  1. Sep 18, 2004 #1
    Hello all

    Let us say we are given a sequence of order 2. By order 2 I mean that we have a sequence in which the differences between the terms forms a sequence of order 1, which has a constant difference between terms. How can I prove that the nth term of a sequence of order 2 can be represented as:

    an^2 + bn + c?


    Or more generally how would I prove that that the nth term of a sequence of order k can be represented as:

    an^k + bn^k-1 +.... + pn + q?

    Any help would we greatly appreciated.

    Thanks
     
  2. jcsd
  3. Sep 18, 2004 #2

    Tide

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    I think you only need to focus on the highest order term. Just look at the difference [itex]P(n+1) - P(n)[/itex] where P(n) is your sequence and use the first term in the binomial expansion of [itex]P(n+1)[/itex].
     
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